9x^2+31x+20=0

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Solution for 9x^2+31x+20=0 equation:


Simplifying
9x2 + 31x + 20 = 0

Reorder the terms:
20 + 31x + 9x2 = 0

Solving
20 + 31x + 9x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
9 the coefficient of the squared term: 

Divide each side by '9'.
2.222222222 + 3.444444444x + x2 = 0

Move the constant term to the right:

Add '-2.222222222' to each side of the equation.
2.222222222 + 3.444444444x + -2.222222222 + x2 = 0 + -2.222222222

Reorder the terms:
2.222222222 + -2.222222222 + 3.444444444x + x2 = 0 + -2.222222222

Combine like terms: 2.222222222 + -2.222222222 = 0.000000000
0.000000000 + 3.444444444x + x2 = 0 + -2.222222222
3.444444444x + x2 = 0 + -2.222222222

Combine like terms: 0 + -2.222222222 = -2.222222222
3.444444444x + x2 = -2.222222222

The x term is 3.444444444x.  Take half its coefficient (1.722222222).
Square it (2.966049382) and add it to both sides.

Add '2.966049382' to each side of the equation.
3.444444444x + 2.966049382 + x2 = -2.222222222 + 2.966049382

Reorder the terms:
2.966049382 + 3.444444444x + x2 = -2.222222222 + 2.966049382

Combine like terms: -2.222222222 + 2.966049382 = 0.74382716
2.966049382 + 3.444444444x + x2 = 0.74382716

Factor a perfect square on the left side:
(x + 1.722222222)(x + 1.722222222) = 0.74382716

Calculate the square root of the right side: 0.86245415

Break this problem into two subproblems by setting 
(x + 1.722222222) equal to 0.86245415 and -0.86245415.

Subproblem 1

x + 1.722222222 = 0.86245415 Simplifying x + 1.722222222 = 0.86245415 Reorder the terms: 1.722222222 + x = 0.86245415 Solving 1.722222222 + x = 0.86245415 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.722222222' to each side of the equation. 1.722222222 + -1.722222222 + x = 0.86245415 + -1.722222222 Combine like terms: 1.722222222 + -1.722222222 = 0.000000000 0.000000000 + x = 0.86245415 + -1.722222222 x = 0.86245415 + -1.722222222 Combine like terms: 0.86245415 + -1.722222222 = -0.859768072 x = -0.859768072 Simplifying x = -0.859768072

Subproblem 2

x + 1.722222222 = -0.86245415 Simplifying x + 1.722222222 = -0.86245415 Reorder the terms: 1.722222222 + x = -0.86245415 Solving 1.722222222 + x = -0.86245415 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.722222222' to each side of the equation. 1.722222222 + -1.722222222 + x = -0.86245415 + -1.722222222 Combine like terms: 1.722222222 + -1.722222222 = 0.000000000 0.000000000 + x = -0.86245415 + -1.722222222 x = -0.86245415 + -1.722222222 Combine like terms: -0.86245415 + -1.722222222 = -2.584676372 x = -2.584676372 Simplifying x = -2.584676372

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.859768072, -2.584676372}

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